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On Parameter Differentiation for Integral Representations of Associated Legendre Functions
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- Фізико-технічні та математичні науки
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- Відділення математики
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2011, том 7
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On Parameter Differentiation for Integral Representations of Associated Legendre Functions
Інший ID:
2010 Mathematics Subject Classification: 31B05; 31B10; 33B10; 33B15; 33C05; 33C10
DOI:10.3842/SIGMA.2011.050
DOI:10.3842/SIGMA.2011.050
Посилання:
On Parameter Differentiation for Integral Representations of Associated Legendre Functions / H.S. Cohl // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 24 назв. — англ.
Підтримка:
This paper is a contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S⁴)”. The full collection is available at http://www.emis.de/journals/SIGMA/S4.html.
I would like to thank Dr. A.F.M. ter Elst for extremely valuable discussions and acknowledge funding for time to write this paper from the Dean of the Faculty of Science at the University of Auckland in the form of a three month stipend to enhance University of Auckland 2012 PBRF Performance. I would like to express my gratitude to the anonymous referees whose helpful comments improved this paper. I would also like to thank F.W.J. Olver for helpful discussions. Part of this work was conducted while the author was a National Research Council Research Postdoctoral Associate in the Information Technology Laboratory of the National Institute of Standards and Technology.
Дата:
2011
Переглядів:
515
Завантажень:
191
Анотація:
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function f: C\{−1,1}→C given by f(z)=z/(√(z+1)√(z−1)).
